Find all values of $ x $where $ f\left(x\right) < 0 $.XXXYYY-11-11-11-1-1-1-6-6-6

$ x > -1 $ and also $ x > -11 $

Find X Values Where f(x) Is Negative: Graph Analysis

Q: How do I know which parts of the graph are negative?

Look for sections where the curve is below the x-axis. The graph shows the function dips below zero to the left of x = -11 and to the right of x = -1.

Q: What do the roots at -11, -6, and -1 mean?

These are x-intercepts where f(x) = 0. They divide the graph into intervals. The root at -6 is where the curve touches the x-axis between the other two roots.

Q: How do I write the final answer correctly?

Use or to connect separate intervals: $ x or \( x > -1 $. Don't use 'and' because x cannot be in both intervals simultaneously!

Q: Should I include the roots in my answer?

No! Since we want f(x) not included. Use symbols, not ≤ or ≥.

Inequality Analysis with Graphical Interpretation

Find all values of $x$

where $f\left(x\right) < 0$ .

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find all values of $x$

where $f\left(x\right) < 0$ .

Step-by-step solution

To determine where the function $f(x)$ is less than 0, observe the graphical representation:

The roots are located at $x = -11$ , $x = -6$ , and $x = -1$ . These are the x-values where the function intersects the x-axis.
Considering the general behavior of quadratic functions, the function is negative between the outer roots unless it passes through below x-axis at multiple roots due to shape.

The given graph suggests the function dips below the x-axis between $x = -11$ and $x = -1$ , passing through $x = -6$ .

After analyzing the intervals:

The interval to the left: $x < -11$
The interval to the right: $x > -1$

Therefore, values of $x$ for which the function $f(x)$ is less than 0 are $x > -1$ or $x < -11$ .

The correct choice is: $x > -1$ or $x < -11$

Final Answer

$x > -1$ or $x < -11$

Key Points to Remember

Essential concepts to master this topic

Graph Reading: Function is negative when curve lies below x-axis
Technique: Identify roots at x = -11, -6, -1 and test intervals
Check: Verify endpoints are excluded since f(x) < 0, not ≤ 0 ✓

Common Mistakes

Avoid these frequent errors

Reading between consecutive roots instead of correct intervals
Don't assume f(x) < 0 between -11 and -1 just because those are outer roots = wrong interval selection! The graph shows the function is actually positive in that middle region. Always trace the curve carefully to see where it dips below the x-axis.

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below intersects the X-axis at points A and B.

The vertex of the parabola is marked at point C.

Find all values of $$ x $$ where $f\left(x\right) > 0$ .

FAQ

Everything you need to know about this question

How do I know which parts of the graph are negative?

Look for sections where the curve is below the x-axis. The graph shows the function dips below zero to the left of x = -11 and to the right of x = -1.

Why isn't the answer just between -11 and -1?

That's a common mistake! The graph actually shows the function is positive in the middle section between the roots. You need to look at where the curve actually goes below the x-axis.

What do the roots at -11, -6, and -1 mean?

These are x-intercepts where f(x) = 0. They divide the graph into intervals. The root at -6 is where the curve touches the x-axis between the other two roots.

How do I write the final answer correctly?

Use or to connect separate intervals: $x < -11$ or $x > -1$ . Don't use 'and' because x cannot be in both intervals simultaneously!

Should I include the roots in my answer?

No! Since we want f(x) < 0 (strictly less than), the roots where f(x) = 0 are not included. Use < or > symbols, not ≤ or ≥.

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